A wake-up call: information contagion and speculative ...

A wake-up call: information contagion and speculative

currency attacks∗

Toni Ahnert† and Christoph Bertsch‡

This version: June 2013[ Link to the latest version ]

Abstract

A successful speculative attack against one currency is a wake-up call for

speculators elsewhere. Currency speculators have an incentive to acquire costly

information about exposures across countries to infer whether their monetary

authority’s ability to defend its currency is weakened. Information acquisition

per se increases the likelihood of speculative currency attacks via heightened

strategic uncertainty among speculators. Contagion occurs even if speculators

learn that there is no exposure. Our new contagion mechanism offers a compelling

explanation for the 1997 Asian currency crisis and the 1998 Russian crisis, both of

which spread across countries with seemingly unrelated fundamentals and limited

interconnectedness.

Keywords: contagion, coordination failure, information acquisition, specula-

tive currency attacks

JEL classification: C7, D82, F31, G01

∗The authors wish to thank Jose Berrospide, Elena Carletti, Christian Castro, Amil Dasgupta, DouglasGale, Piero Gottardi, Antonio Guarino, Todd Keister, Ralf Meisenzahl, Morten Ravn, David Rahman,Wolfram Richter, Jean-Charles Rochet, Nikita Roketskiy, Myung Seo, Michal Szkup, and Dimitri Vayanosas well as seminar participants at the Federal Reserve Board of Governors, University College London,European University Institute, and TU Dortmund for fruitful discussions and comments. An earlier versionof the paper was circulated under the title ”A wake-up call: contagion through alertness” (June 2012).†London School of Economics and Political Science, Financial Markets Group and Department of Eco-

nomics, Houghton Street, London WC2A 2AE, United Kingdom. Part of this research was conducted whenthe author was visiting the Department of Economics at New York University and the Federal Reserve Boardof Governors. Email: [email protected].‡Department of Economics, University College London, Gower Street, London WC1E 6BT, United King-

dom. Email: [email protected].

1 Introduction

Financial contagion can happen even if countries have seemingly unrelated fundamentals

and limited interconnectedness. A prominent example is Brazil that got affected by the 1998

Russian crisis although Brazil’s exposure to Russia was very limited. Our paper is motivated

by this phenomenon and provides a novel contagion mechanism in coordination games that

does not rely on common exposures and interconnectedness. It explains why a contagious

spread of a crisis can occur even if agents learn that their country’s fundamentals are not

exposed to crisis events elsewhere.

We define contagion as an increase in the likelihood of a financial crisis in one country

after another country has been affected by a financial crisis. Our new contagion mechanism

is developed in an incomplete information game of speculative currency attacks, based on

Morris and Shin [29, 30] and following the tradition of the global games literature. The

main finding of our paper is that contagion occurs even if agents get informed and learn

that their country is not exposed to a crisis event elsewhere. But what is more, we find

that the scenario where agents learn good news about their country’s fundamentals can be

associated with a higher likelihood of financial crises relative to the scenario where agents

learn no news and stay uninformed about the exposure. At first glance this second result

may be surprising. However, the underlying mechanics are intuitive. Key is that learning

the news of no exposure can lead to more financial fragility if the news of no exposure is not

only associated with a more favourable public information, but also affects the information

precision of speculators.

We demonstrate that the above described contagion effect prevails as an equilibrium

phenomenon if learning is endogenous. Furthermore, endogenous information helps us to cap-

ture the idea of contagion-through-alertness. Observing a ’trigger event’ in another country

or region such as a banking crisis, a balance-of-payments crisis or a sovereign debt crisis is

a wake-up call for a domestic investor and makes her alert. Taking the example of spec-

ulative currency attacks, a successful speculative currency attack against one county is a

wake-up call for currency speculators elsewhere. Speculators wonder whether their country’s

fundamentals are affected and, hence, the ability of their monetary authority to defend its

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currency is weakened. While it is ex-ante unknown whether fundamentals are correlated

across countries, there is some chance of a positive correlation. For that reason speculators

expect that their monetary authority’s ability to defend its currency can be detrimentally

affected. This may be due to macroeconomic factors such as common shocks or due to inter-

connectedness and institutional similarities of their country with the ”ground zero country”,

which was attacked initially.1 Consequently, currency speculators wish to determine the

extent of their exposure to the trigger event in the ground zero country by acquiring costly

information. We call information acquisition after such a trigger event elsewhere an alertness

effect and demonstrate that it can cause contagion. Interestingly, when currency speculators

learn that there is no correlation, financial fragility can be higher than without information

acquisition. Thus, fragility in one country can lead to fragility in a second country although

fundamentals are independent – a contagion-through-alertness effect. The fragility in the

second country is a direct consequence of the change in the information precision of spec-

ulators due to learning. It arises in the context of coordination problems. This is the case

because additional information about the cross-country correlation of fundamentals has two

effects in our incomplete information game.

Mean effect Having observed a successful currency attack due to a low realisation of

fundamentals in the other country, a speculator’s posterior mean about her country’s funda-

mentals improves upon learning that there is no cross-country correlation. This is because

the low fundamental realisation in the other country shows to be irrelevant for her coun-

try’s ability to defend its currency. The mean effect is associated with a lower likelihood of

successful speculative currency attacks after learning that fundamentals are uncorrelated.2

1In practise an exposure may arise due to trade-links, financial links or institutional similarities. Both,macroeconomic and financial similarities show to play an important role. In early empirical work Glick andRose [19] find that ”currency crises tend to be regional” (page 603) and underline geographic proximity asan important factor. Instead Van Rijckeghem and Weder [39, 40] find that for the most recent episodes ofcurrency crises spillovers through bank lending played a more important role. Finally, Dasgupta et. al. [14]find that institutional similarity to the ”ground zero country” is an important determinant for the directionof financial contagion.

2A good description of the mean effect can be found in Vives [41].

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Variance effect The information about the cross-country correlation of fundamentals also

affects the information precision. In particular, the relative precision of public information is

lowest if speculators learn that fundamentals are uncorrelated. We find that a lower relative

precision of public information increases (decreases) the likelihood of successful speculative

currency attacks if the prior belief is that fundamentals are strong (weak). In other words, the

result depends on whether the equilibrium fundamental threshold is above or below the public

information of speculators.3 The reason being, that a lower relative precision of public signals

increases strategic uncertainty – the variance effect. This increase in strategic uncertainty is

reflected in a more dispersed belief about other speculators’ posterior. Given a prior belief

that fundamentals are strong (weak), the increase in strategic uncertainty makes speculators

more concerned about other speculators receiving a bad (good) private signal. The shift in

beliefs about other speculators’ posterior induces more (less) aggressive speculative currency

attacks. As a result, the variance effect can be associated with an increase or a decrease in

the likelihood of successful currency attacks when comparing the scenario where speculators

learn that fundamentals are uncorrelated relative to scenario where speculators do not learn

about the correlation (and, hence, expect a potentially positive cross-country correlation).

The direction of the variance effect crucially depends on whether the prior belief is that

fundamentals are strong or weak.

In sum, the mean effect and the variance effect go in opposite directions given a prior

belief that fundamentals are strong (which implies a large degree of coordination failure).

Having observed a successful currency attack due to a low realisation of fundamentals in one

country, the news of no cross-country correlation implies more fragility in another country

through heightened strategic uncertainty if the variance effect dominates the mean effect.

Contagion can occur even if agents learn that they are not exposed to the crisis event

elsewhere which triggered learning in the first place.

The novel contagion mechanism prevails as an equilibrium phenomenon with endoge-

nous information acquisition about the cross-country correlation. This is because currency

3Similar results have been discussed in a global-games model by Metz [27], and by Rochet and Vives[36]. He and Xiong [21] provide an alternative framework in which they establish a ”volatility effect”. Theirvolatility effect is to some degree related to the variance effect, but is not based on a change in strategicuncertainty.

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speculators have an ex-ante incentive to acquire information on the cross-country correla-

tion whenever the cost of information is sufficiently low. Intuitively, the information on the

cross-country correlation of fundamentals helps a speculator to improve her forecast about

her country’s fundamental as well as the behaviour of other agents. We demonstrate that

a speculator can earn a higher gross expected payoff after adjusting her attack strategy be-

cause of being informed. In particular, an informed speculator obtains a higher expected

payoff than an uninformed speculator by acting more (less) aggressively after receiving in-

formation that lowers (improves) her forecast for fundamentals. By doing so, an informed

speculator increases her expected benefits (reduces her expected costs) from participating in

a successful (unsuccessful) currency attack when fundamentals are weak (strong).

Literature Our paper is related to Morris and Shin [29, 30] who develop an incomplete

information game of speculative currency attacks in the tradition of the global games liter-

ature pioneered by Carlsson and van Damme [8]. We differ in two main aspects. First, we

consider a two-country model with potentially correlated fundamentals and address the issue

of contagion across countries. Currency speculators move sequentially such that speculators

in the second country decide whether to attack their country’s currency after observing the

outcome in the first country (wake-up call). Second, speculators in the second country can

acquire information about the cross-country correlation of fundamentals (alertness effect).

Similar to Corsetti et. al. [10] speculators can be asymmetrically informed, but our focus is

on contagion.4

There is a large existing literature on contagion in financial economics and in interna-

tional finance.5 With few exceptions the existing literature relies on common exposures and

interconnectedness.6 We demonstrate with our new contagion mechanism that contagion

4The authors also consider different sizes of currency speculators and its effect on the likelihood andseverity of a currency crisis. Instead our speculators are of equal size as our contagion-through-alertnessmechanism does not require signalling or herding.

5An excellent recent literature survey can be found in Forbes [18] and a more detailed description of therelation of our paper to the literature on contagion is given in section 4.

6Similar to us Goldstein and Pauzner [20] do not rely on correlated fundamentals. The authors obtaincontagion because of risk averse speculators who are invested in two countries. After a crisis in the firstcountry speculators become more averse to strategic risk and have a larger incentive to withdraw theirinvestment.

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can occur in the absence of interconnectedness or common exposures. This allows us to

offer an explanation for the occurrence of contagious currency or banking crises even if the

fundamentals of the affected countries or financial institutions are seemingly unrelated and

even if there is only limited interconnectedness. Such a situation does not only apply to the

aforementioned example of Brazil, which was affected by the 1998 Russian financial crisis,7

but it is also relevant for the Asian balance-of-payment and banking crisis in 1997. In Asia

it was at least for some of the affected countries the case that the spread of contagion is

difficult to explain without leaning on models with multiple equilibria and the possibility of

sudden unexplained shifts in market confidence (see Radelet and Sachs [35] and Krugman

[25]).

The specialty of our new contagion mechanism is that contagion can occur even if spec-

ulators learn the good news that there is no cross-country correlation. What is more, the

phenomenon of a higher likelihood of currency attacks after learning that there is zero corre-

lation can be the consequence of ex-ante optimal information acquisition. In complementary

subsequent work, Ahnert [2] examines the amplification of the probability of bank runs or

sovereign debt crises via endogenous acquisition of private information after learning bad

news. Moreover, he investigates the strategic aspects of information acquisition choices and

equilibrium multiplicity. By contrast, we examine learning about the stochastic exposure to

a crisis country and demonstrate how contagion can arise even after good news.

Our new contagion mechanism is general and lends itself to several applications. It

applies to coordination problems in which the payoff from acting depends on both, the

underlying state of the world and the proportion of other agents acting. In the example of

bank runs, the trigger event is that bank creditors of one bank observe a run on another bank.

In the Arab spring, political activists in one country observe a revolution in a neighbouring

country and decide whether or not to attempt a revolution themselves (see Edmond [15]).

Alternative applications are sovereign debt crises or foreign direct investment across emerging

markets (see Dasgupta [13]). Common to these examples is that agents are not directly

7The spread of the 1998 Russian financial crisis to Brazil cannot be attributed to fundamental reasonsor cross-country linkages on which most of the international finance literature on contagion relies. AlthoughBrazil had a very limited exposure, it was still one of the most affected countries (see Bordo and Murshid[6] and Pavlova and Rigobon [34]).

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affected by the trigger event but might be affected indirectly.

This paper is organised as follows. The model is described in section 2 and solved in

section 3. Our main focus is on establishing the novel contagion mechanism with exogenous

information. In section 3.5 we extend our model and show that our contagion mechanism can

be an equilibrium phenomenon with endogenous information acquisition. A more detailed

discussion of the related literature is offered in section 4. Finally, section 5 concludes. All

proofs and most derivations are relegated to the appendices.

2 Model

The economy extends over two dates t ∈ {1, 2} and consists of two countries, where the first

(second) country only moves at the first (second) date. Each country is inhabited by a unit

continuum of risk-neutral agents interpreted as currency speculators indexed by i ∈ [0, 1].8

A country is characterised by its fundamental θt that measures the difficulty of a successful

currency attack from the perspective of speculators. For example, a country’s fundamental

represents the government’s strength to defend its currency with its foreign reserves (e.g

Morris and Shin [29]).

Speculators play a simultaneous-move game with binary action space ait ∈ {0, 1}: each

speculator either attacks the currency (ai = 1) or does not attack (ai = 0). The success

of a currency attack depends on both the fundamental θt and the proportion of attacking

speculators denoted by At ≡∫ 1

0ait di. A speculative attack is successful if the fraction of

acting speculators weakly exceeds the strength of the fundamental (At ≥ θt). The benefit

of a speculator from participating in a successful currency attack is given by b > 0. Her

loss from participating in an unsuccessful currency attack is given by l > 0. As in Vives

[41], the payoff from not attacking is constant for simplicity and normalised to zero, i.e.

8Currency speculators may act nationally or internationally. While speculators can be identical acrosscountries, we only require the sequential timing of events.

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u(ait = 0, At, θt) = 0:

u(ait = 1, At, θt) =

b if At ≥ θt

−l if At < θt

(1)

A key feature of our model is the initial uncertainty about the correlation between

fundamentals across countries denoted by ρ ≡ corr(θ1, θ2). The correlation follows a bivariate

distribution and is zero with probability p ∈ (0, 1) and takes a positive value ρH ∈ (0, 1)

with probability 1− p:

ρ =

ρH w.p. 1− p

0 w.p. p

(2)

Fundamentals in both countries are conditionally distributed as a bivariate normal with

mean µt ≡ µ, precision αt ≡ α > 0, and correlation ρ.9 Speculators in country 2 observe

whether a currency attack in country 1 was successful. If an attack was successful, speculators

also observe the realisation of θ1.10 The bounds on the positive correlation ρH ensure that

a speculator in region 2 who observes the realisation θ1 in region 1 and learns that the

correlation takes the positive value has more precise information but is still imperfectly

informed.

As in the global games literature pioneered by Carlson and van Damme [8], each

speculator receives a private signal xit about her country’s fundamental before deciding

whether to attack:

xit ≡ θt + εit (3)

where the idiosyncratic noise εit is identically and independently normally distributed across

speculators and countries with zero mean and precision γ > 0. All distributions are common

knowledge.

The game in country 2 has two stages. In stage 2 speculators play the speculative

9If information about the fundamental is complete, then multiple equilibria arise for θt ∈ (0, 1). Someauthors therefore restrict attention to µ ∈ (0, 1) to match this parameter restrictions in the incompleteinformation setup.

10The motivation for this assumption is as follows. After a successful currency attack it becomes publicinformation why the monetary authority in country 1 was too weak to defend its currency. In contrast, if thespeculative currency attack in country 1 is unsuccessful, then the actual strength of country 1’s monetaryauthority θ1 remains unknown.

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attack coordination game. If information is exogenous, a proportion of speculators n ∈ [0, 1]

learn the correlation between fundamentals and n is common knowledge. We derive our

main result in this setup. As an extension, we consider endogenous information acquisition

such that speculators play an information acquisition game in stage 1. Each speculator may

simultaneously purchase a signal about the correlation of the fundamentals at a cost c > 0.

The signal is common to all speculators and publicly available. A figurative example of such

a signal is a newspaper, which takes money to buy and time to absorb. In terms of whole-

sale investors or currency speculators, it could be the access to Bloomberg and Datastream

terminals or for the hiring of analysts to interpret the publicly available information. Infor-

mation acquisition is costly in each case. The signal about the correlation of fundamentals

is perfectly revealing for simplicity again.

The model is summarised in the following timeline:

Date t = 1

• ρ is drawn. Then θ1 and θ2 are drawn from a bivariate normal with correlation ρ.

• Speculators in country 1 receive their private signals xi1 and decide simultaneously

whether to attack the currency.

• Payoffs are realised. The fundamental θ1 is publicly observed by speculators in both

countries after a successful speculative currency attack.

Date t = 2

• Stage 1:

– Exogenous information: a known proportion of speculators n ∈ [0, 1] learn the

realisation of the correlation.

– Endogenous information: speculators in country 2 simultaneously decide whether

to purchase a publicly available signal about the correlation of fundamentals ρ at

cost c > 0.

• Stage 2:

– Speculators in country 2 receive their private signals xi2 and decide simultaneously

whether to attack the currency.

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– Payoffs are realised.

3 Equilibrium

The focus of this paper is on the equilibrium in country 2. In particular, we describe

how events in country 1 influence this equilibrium, contrasting the situation of known and

unknown correlation between fundamentals. It is therefore useful to revise briefly the equi-

librium in country 1, which is a standard coordination game as in e.g. Vives [41].

3.1 Country 1

Observe that the differential payoff u(ai1 = 1, A1, θ1)− u(ai1 = 0, A1, θ1) is increasing in A1

and decreasing in θ1.11 Denote with φ(x) the pdf of a normal distribution. We can define a

symmetric Bayesian equilibrium as follows.

Definition 1 An equilibrium in country 1 is a speculative attack decision a(xi1) and a an

aggregate mass of attackers A1 ≡ A(θ1), such that:

a(xi1) ∈ arg maxai1∈{0,1}

E[u(ai1, A1, θ1)|xi1] (4)

A(θ1) =

∫ +∞

−∞a(xi1)

√γφ(√γ(xi1 − θ1))dxi1 (5)

We consider monotone equilibria, i.e. equilibria where there exists an individual private

signal threshold x∗1 and a fundamental threshold θ∗1 such that (a) an individual speculator

only attacks if xi1 ≤ x∗1 and (b) a speculative currency attack is successful if and only if

θ1 ≤ θ∗1.

In equilibrium two conditions have to be satisfied. First, the critical fraction of attack-

ing speculators A(θ∗1) has to equal the critical fundamental threshold above which it pays

11Notice that it is (not) a dominate strategy to attack the currency if θt ≤ 0 (θt ≥ 1). Instead fundamentalsare ”critical” in the intermediate range θt ∈ (0, 1). Here multiple equilibria can be sustained by self-fulfillingexpectations if the realisation of θ1 is common knowledge.

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to attack. Second, a speculator with the threshold signal x∗1 has to be indifferent between

attacking and not attacking the currency given θ∗1. These two equilibrium conditions can be

combined to one equation which implicitly defines θ∗1:12

F1(θ∗1) ≡ Φ

(α√α + γ

(θ∗1 − µ)−√γ

√α + γ

Φ−1(θ∗1)

)=

l

b+ l(6)

It can be shown that there exists a unique θ∗1 solving equation (6) if α√γ<√

2π. Hence, there

exists a unique Bayesian equilibrium in threshold strategies whenever the relative precision of

the private signal is sufficiently high. Following Morris and Shin [30] we can use an iterated

dominance argument to show that there do not exit non-monotone equilibria, meaning that

the above equilibrium is in fact unique.

If γ is sufficiently high, then θ∗1 is decreasing in µ and in lb+l

. Hence, there are two

possible rankings of the equilibrium thresholds depending on the belief about the prior mean

of the fundamental. First, the prior belief about fundamentals is said to be weak when

µ ∈ (0, 1) takes on a low value, while the relative cost of an unsuccessful attack lb+l

is low,

i.e. if:√γΦ−1(µ) +

√α + γΦ−1

(l

b+ l

)< 0 (7)

A prior belief that fundamentals are weak leads to strong attacks on the currency: 0 < µ <

θ∗1 < 1, implying little coordination failure. Second, the prior belief about fundamentals is

said to be strong when the above inequality is reversed, meaning that µ takes on a high value,

while the relative cost of an unsuccessful attack lb+l

is high. A strong prior on fundamentals

leads to less frequent currency attacks: 0 < θ∗1 < µ, implying a large degree of coordination

failure. These rankings are for finite private noise (γ < ∞); refer to Appendix A.1.2 for a

detailed examination.

While the equilibrium analysis in country 1 is standard, the analysis in the next three

sections contains the main contribution of our paper. For the remainder we can abstract

from all the details of the equilibrium in country 1 and consider the equilibrium in country

2 as a function of the realisation of the observed fundamental θ1. In other words, we treat

12See Appendix section A.1.1 for details.

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θ1 as a public signal and focus on the equilibrium in country 2. We are interested in the

case where the public signal is low, i.e. θ1 < µ, and associated with a successful speculative

attack in country 1.13 The aim is to analyse how country 2 is affected after observing a

country-1 fundamental realisation below average.

3.2 Country 2: Symmetrically informed speculators

To demonstrate the core mechanics of our novel contagion mechanism endogenous infor-

mation acquisition is not necessary. Furthermore, we also do not need to allow for asym-

metrically informed speculators. For that reason we abstract in this section from the first

stage of the game at date t = 2 and analyse a simplified model with symmetrically informed

speculators where either everybody or nobody can observe the publicly available signal on

the cross-country correlation, i.e. the ”polar cases” n = 1 and n = 0, respectively. In the

discussion of the polar case when all speculators are informed (n = 1) we can already un-

cover the mean effect and the variance effect, which are at the core of our novel contagion

mechanism. Based on this analysis, we establish our novel contagion mechanism in section

3.3.

However, we emphasise that endogenous information acquisition is an interesting part

of the contagion-through-alertness effect developed in this paper. For that reason we extend

the analysis of section 3.2 by introducing the information acquisition stage at the beginning of

date t = 2 in section 3.5. Here we allow for asymmetrically informed speculators (0 < n < 1)

and demonstrate how endogenous information acquisition can be triggered by a wake-up

call event that makes speculators alert. Most importantly, for a sufficiently low cost of

information on ρ, there exists a unique equilibrium where all speculators want to be informed

and where there is a contagious spread of speculative currency attacks even if speculators

learn that they are not exposed to θ1.

13Notice that we can guarantee that any fundamental realisation θ1 < µ implies that speculative currencyattacks are successful whenever the relative cost of attacking in country 1 is sufficiently low. A numericalexample is provided when we establish the novel contagion mechanism in section 3.3.

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Notation Let’s introduce some notation to distinguish informed and uninformed specula-

tors. Using the subscripts I and U , let ai2I (ai2U) denote the action of informed (uninformed)

speculator i in country 2 and let A2I (A2U) denote the aggregate proportion of attacking

speculators in country 2 that are informed (uninformed).

Informed speculators The belief of an informed speculator about the mean of θ2 depends

on ρ. Let’s denote the conditional mean with µ′2(ρ, θ1) ≡ ρθ1 + (1 − ρ)µ2. An informed

speculator who learns that ρ = 0 beliefs that θ2 is distributed according to:

θ2 ∼ N(µ′2(0, θ1),

1

α

), (8)

where µ′2(0, θ1) = µ2 = µ. Instead if she learns that ρ = ρH , then she believes θ2 follows:

θ2|θ1 ∼ N(µ′2(ρH , θ1),

1− ρ2H

α

), (9)

where µ′2(ρH , θ1) = ρHθ1 + (1− ρH)µ. We can see that the mean of the belief shifts towards

θ1 and the precision unambiguously increases, the larger ρH .

Uninformed speculators An uninformed speculator observes θ1 and uses the prior dis-

tribution about the correlation ρ to update her belief about the fundamental in country 2.

As a result, the belief about the distribution of θ2 is a mixture distribution. Before receiving

her private signal, the uninformed speculator believes that θ2 is drawn with probability p

from the normal distribution described in equation (8) and with probability 1− p from the

normal distribution described in equation (9).

Figure 1 depicts the beliefs about the distribution of θ2 for informed speculators given

in equations (8) and (9) as brown dashed and blue dotted lines, respectively. The uninformed

speculators’ belief about the distribution of θ2 is described by the red solid line and denoted

with θ2|U , where U stands for uninformed.14 Informed speculators who learn that there is

no cross-country correlation, i.e. ρ = 0, have a belief with the highest mean and dispersion,

14For the chosen parameters (high p and not too small θ1) the mixture distribution is not bimodal andrelatively close to a normal distribution.

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while informed speculators who learn that ρ = ρH have a belief with the lowest mean and

dispersion.

0 0.5 1

Θ2Θ2ÈΘ1

Θ2ÈU

Μ=0.9Θ1=0

Figure 1: Highest after learning that ρ = 0. Parameters used: p = 0.7, α = 0.7, ρH = 0.5and θ1 = 0.

We continue in section 3.2.1 by discussing the polar case n = 1 where all speculators

are informed. In section 3.2.2 we analyse the role of public information and information

precision, which lays the foundation for our novel contagion mechanism. Then we shift in

section 3.2.3 focus to the analysis of the problem faced by uninformed speculators who do

not learn the realisation of ρ. Finally, we examine the equilibrium for the polar case n = 0

where all speculators are uninformed in section 3.2.4.

3.2.1 Equilibrium for the special case n = 1: classical information contagion

The special case of completely informed speculators captures the classical information con-

tagion channel, which is distinct from our novel contagion mechanism to be established in

section 3.3. A low fundamental realisation in country 1 constitutes ”bad news” for the fun-

damentals in country 2 if speculators learn that fundamentals are positively correlated, i.e.

ρ = ρH . The strength of this effect is measured by ρH .15

15An example of this information contagion channel is Acharya and Yorulmazer [1] who show that fundingcosts of one bank increase after bad news about another bank if the banks’ loan portfolio returns have acommon factor.

13

We show that there exists again a unique equilibrium in threshold strategies as in our

analysis of country 1. The sufficient condition on the relative precision of private information

is given byα

1−ρH√γ<√

2π. A successful currency attack occurs in country 2 if its fundamental

realisation is below its unique equilibrium threshold, i.e. θ2 ≤ θ∗2I,ρ ∈ (0, 1)), where the

subscript I stands for informed. The critical threshold for the country’s fundamental θ∗2I,ρ

depends on ρ and is implicitly defined by:16

F2(θ∗2I,ρ, ρ) ≡ Φ

( α1−ρ2 (θ∗2I,ρ − [ρθ1 + (1− ρ)µ])−√γΦ−1(θ∗2I,ρ)√

α+(1−ρ2)γ(1−ρ2)

)=

l

b+ l(10)

where ρ = 0 (ρ = ρH) if speculators learn that there is (no) exposure. Again the left-

hand side is monotone and decreasing in θ∗2I,ρ for a sufficiently high relative precision of

the private signal. If ρ = ρH , the left-hand side of equation (10) is decreasing in θ1 and

we can conclude that dθ∗2I,ρH/dθ1 < 0. A lower observed fundamental in country 1 implies

that the fundamental in country 2 is likely to be low as well if ρ = ρH . Speculators expect

little defence by the country-2 government against a currency attack. Consequently, it is

optimal for speculators to attack the currency in country 2 more aggressively, thus raising the

fundamental equilibrium threshold of country 2 below which a currency attack is successful.

3.2.2 The role of public information and information precision

The aim of this section is to shed light on the interplay between the mean effect and the

variance effect, which crucially influences the ordering of equilibrium thresholds θ∗2I,0 and

θ∗2I,ρH in the two states of the world. This interplay between the mean effect and the variance

effect will serve as a basis for the novel contagion mechanism developed in section 3.3.

We find that the mean effect increases θ∗2I,0 relative to θ∗2I,ρH , while the variance effect

tends to decrease (increase) θ∗2I,0 relative to θ∗2I,ρH if the prior belief is that fundamentals are

strong (weak). As a result, we can only have that θ∗2I,0 > θ∗2I,ρH if the prior belief is that

16See Appendix A.3 for a detailed analysis, where the equilibrium and sufficient conditions for uniquenessare derived.

14

fundamentals are strong and the variance effect outweighs the mean effect. However, it is

important to notice that the ordering of the equilibrium thresholds for different states of the

world should not be confused with the ordering of likelihoods of successful currency attacks

because θ2 is drawn from a different distribution depending on the state of the world.

The prior belief on fundamentals Similar to before, a weak prior belief on fundamentals

leads to strong attacks against the currency independent of the realisation of ρ, i.e. µ <

θ∗2I,0 < 1 and ρHθ1 + (1− ρH)µ < θ∗2I,ρH < 1. Instead, a strong prior belief on fundamentals

leads to weak attacks against the currency independent of the realisation of ρ, i.e. 0 < θ∗2I,0 <

µ and 0 < θ∗2I,ρH < ρHθ1 + (1 − ρH).17 For ”intermediate” values of µ the prior belief on

fundamentals depends on the realisation of ρ. More formally:

Definition 2 The prior belief is that fundamentals are strong independently of the realisation

of ρ if µ′2 ∈ S1, and that fundamentals are weak independently of ρ if µ′2 ∈ S2. Fundamentals

are expected to be strong or weak depending on the realisation of ρ if µ′2 /∈ {S1, S2}.

Where:

S1 =

{{µ′2, θ1, ρH , b, l} : [µ′2 > max{X(0), X(ρH)}]

}(11)

S2 =

{{µ′2, θ1, ρH , b, l} : [µ′2 < min{X(0), X(ρH)}]

}(12)

and:

X(ρ) ≡ Φ

(−

√α

1−ρ2 + γ√γ

Φ−1

(l

b+ l

))− ρ(θ1 − µ). (13)

Mean effect It is well known that more favourable public information, i.e. a higher prior

mean µ, is associated with a lower equilibrium fundamental threshold. In our model not

only a decrease in θ1, but also an increase in ρH are associated with a decrease in the prior

mean. Given that the prior mean is higher if fundamentals are not correlated, i.e. if ρ = 0,

17Recall the discussion for country 1 in section 3.1 and the derivations in Appendix section A.1.2.

15

than if fundamentals are correlated, i.e. if ρ = ρH , we have that the mean effect tends to

lower θ∗2I,0 relative to θ∗2I,ρH .

Variance effect It crucially depends on the prior belief on fundamentals if the equilibrium

fundamental threshold θ∗2I,0 increases or decreases in the precision of the private signal γ and

the public signal α. To our knowledge this was first analysed in detail by Metz [27]. For the

special case b = l = 12, the equilibrium fundamental threshold θ∗2I,0 increases (decreases) in

the precision of the private signal γ when the prior belief is that fundamentals are strong

(weak). This result is consistent with the findings of Rochet and Vives [36]. A related result

is that the above relationship is opposite when considering a change in the precision of the

pubic signal α.

Notice that the precision of the public signal is lower in the state where there is no

correlation (α < α1−ρ2

H). As a consequence, the variance effect tends to increase (decrease)

θ∗2I,0 relative to θ∗2I,ρH if the prior belief is that fundamentals are strong (weak) independent

of the realisation of ρ. For a prior belief that fundamentals are strong there is a clear

tension between the mean and the variance effect, which go in opposite directions. A formal

derivation can be found in Appendix section A.2.1. Here we also discuss the general case for

any b, l > 0, which requires somewhat stronger conditions on µ′2(ρ, θ1). The intuition for the

results is developed in the next paragraph.

Intuition Given a private signal precision γ, a speculator with a prior belief that fun-

damentals are strong who receives a bad signal places the more weight on her bad private

signal, the more dispersed the prior (the smaller α). Other speculators knowing this, believe

that more speculators will have a low posterior that induces them to attack the currency if

α is smaller. They optimally decide to attack the currency more aggressively. To see this

consider the probability that a given informed speculator i (with signal xi2) attaches to the

event that another informed speculator j has a smaller posterior. Denote with Θi2I,0 the

posterior of a given informed speculator i:

Θi2I,0 ≡ θ2|xi2 ∼ N(αµ+ γxi2

α + γ,

1

α + γ

). (14)

16

She believes that another informed speculator j has a smaller posterior with probability:

Pr{Θj2I,0 < Θi2I,0|Θi2I,0} = Φ

(α(α + γ)

α + 2γ[Θi2I,0 − µ)

)(15)

Notice that the equilibrium posterior mean Θ∗i2I,0 is smaller than µ if the prior belief is

that fundamentals are strong. A given speculator i with a low private signal xi2 close to

the equilibrium attack threshold x∗2 expects a larger fraction of other speculators receiving

a signal that corresponds to a lower posterior if α is smaller. This induces speculator i to

optimally respond by attacking more aggressively. As a result, the variance effect tends to

increase θ∗2I,0 relative to θ∗2I,ρH .18

3.2.3 Bayesian updating by uninformed currency speculators

Uninformed speculators do not know the realisation of ρ. However, they use their private

signal xi2 to update their prior belief on the distribution of ρ. In particular, uninformed

speculators use Bayes’ rule to form a belief on the probability that θ2 is not correlated to θ1.

Using Bayes’ rule we can derive Pr{ρ = 0|θ1, xi2} as:

Pr{ρ = 0|θ1, xi2} =Pr{xi2|θ1, ρ = 0} ∗ p

pPr{xi2|θ1, ρ = 0}+ (1− p) Pr{xi2|θ1, ρ = ρH}(16)

18If instead the prior belief is that fundamentals are weak, then there is only a relatively small degree ofcoordination failure. Here the increase in strategic uncertainty caused by a smaller level of α has an oppositeeffect. Now speculators who receive a private signal that contradicts the prior, i.e. a good signal relative tothe low prior mean, play a key role as they place more weight on their good private signal. Other speculatorsknowing this belief that more speculators will have a high posterior that induces them not to attack thecurrency if α is smaller. They optimally decide to attack the currency less aggressively. We have an increasein coordination failure. This tends to decrease θ∗2I,0 relative to θ∗2I,ρH .

17

where:19

Pr{xi2|θ1, ρ = 0} =1√

Var[xi2|θ1, ρ = 0]φ

(xi2 − E[xi2|θ1, ρ = 0]√

Var[xi2|θ1, ρ = 0]

)=

(1

α+

1

γ

)− 12

φ

(xi2 − µ√

1α

+ 1γ

)(17)

Pr{xi2|θ1, ρ = ρH} =1√

Var[xi2|θ1, ρ = ρH ]φ

(xi2 − E[xi2|θ1, ρ = ρH ]√

Var[xi2|θ1, ρ = ρH ]

)=

(1− ρ2

H

α+

1

γ

)− 12

φ

(xi2 − [ρHθ1 + (1− ρH)µ]√

1−ρ2H

α+ 1

γ

)(18)

An examination of Pr{ρ = 0|θ1, xi2} reveals that:

dPr{ρ = 0|θ1, xi2}dθ1

≥ 0 if xi2 ≤ ρHθ1 + (1− ρH)µ

< 0 otherwise.

(19)

We can see that for a relatively low private signal, an increase in θ1 leads speculators to

belief that a zero cross-country correlation of fundamentals is more likely.

Furthermore, we can find that:20

dPr{ρ = 0|θ1, xi2}dxi2

> 0 if θ1 < µ and xi2 ≥ ρHθ1 + (1− ρH)µ

≤ 0 if θ1 ≥ µ and xi2 ≤ ρHθ1 + (1− ρH)µ

Q 0 otherwise.

(20)

The results are intuitive. We are interested in a scenario where speculators in country

2 observe a successful currency attack with a realisation of θ1 smaller than µ as described

in section 3.1. Recall that the prior distribution is more dispersed if ρ = 0. As a result we

19Notice that the variance terms are unconditional on θ2. Hence, we have to compute the sum of Var[εi2]

and the variance of θ2, which is 1α or

1−ρ2Hα .

20See Appendix section A.4 for a derivation.

18

have that an extremely high or low private signal induces uninformed speculators to believe

that the state of the world is very likely to be ρ = 0, i.e. limxi2→+∞ Pr{ρ = 0|θ1, xi2} = 1

and limxi2→−∞ Pr{ρ = 0|θ1, xi2} = 1.

Whenever speculators in country 2 observe a relatively good signal (i.e. xi2 ≥ ρHθ1 +

(1− ρH)µ), while observing a successful currency attack in country 1 (i.e. θ1 < µ given that

fundamentals are strong), an increase in their private signal leads them to belief that cross-

country fundamentals are with a higher probability not correlated. Instead if speculators in

country 2 observe a relatively bad signal (i.e. xi2 < ρHθ1 + (1 − ρH)µ), while observing a

successful currency attack in country 1, then the relationship between Pr{ρ = 0|θ1, xi2} and

xi2 is non-monotone. In case the private signal is low but not too low, we still have that

dPr{ρ=0|θ1,xi2}dxi2

> 0. However, in case the private signal is very low we have that dPr{ρ=0|θ1,xi2}dxi2

≤

0 due to the more dispersed prior distribution if ρ = 0.

3.2.4 Equilibrium for the special case n = 0: how Bayesian updating changes

the analysis

As before we are interested in monotone equilibria. Again two conditions have to be satisfied

in equilibrium. The critical mass condition and the indifference condition. A combination

of both leads to:21

G(θ∗2U , θ1) ≡ Pr{ρ = 0|θ1, x∗2(θ∗2U)}ΦI,ρ=0(θ∗2U)

+ Pr{ρ = ρH |θ1, x∗2(θ∗2U)}ΦII,ρ=ρH (θ∗2U , θ1) =

l

b+ l(21)

where:

ΦI,ρ=0(θ∗2U) ≡ Φ

(α√α + γ

(θ∗2U − µ)−√γ

√α + γ

Φ−1(θ∗2U)

)ΦII,ρ=ρH (θ∗2U , θ1) ≡ Φ

(δ(ρH)(θ∗2U − [ρHθ1 + (1− ρH)µ])−

√γ√

α1−ρ2

H+ γ

Φ−1(θ∗2U)

)21See Appendix section A.5.1 for details.

19

and δ(ρH) ≡ α1−ρ2

H/√

α1−ρ2

H+ γ. Recall that the subscript U stands for uninformed. G(θ∗2U , θ1)

looks like a mixture of F2(θ∗2I,0, ρ = 0) and F2(θ∗2I,ρH , ρ = ρH). But now there is only one fun-

damental threshold θ∗2U for both states of the world, as uninformed speculators use the same

strategies in both states. Different to before G(θ∗2U , θ1) is now harder to characterise due to

the dependency of the weights on the private signal. Is our focus on monotone equilibrium

still justified?

First, we can prove that Pr{θ2 ≤ θ∗2U |θ1, xi2} is monotonically decreasing in xi2 using

the result of Milgrom [28]. This is true although the probability weights in the indifference

condition are non-monotone in xi2. Refer to Appendix section A.5.2 for the derivation. The

essentially same argument is used in Chen et. al. [9].22

Furthermore, let us do the thought experiment and analyse the best-response of a given

speculator if varying the critical attack threshold used by other speculators. Letting ˆθ2U(x2)

be the critical fundamental threshold when players other than i use a threshold strategy with

the critical attack threshold x2, we can show that Pr{θ2 ≤ ˆθ2U(x2)|θ1, xi2} is increasing in x2.

The best response of a player i is to use a threshold strategy with critical attack threshold

xi2, where Pr{θ2 ≤ ˆθ2U(x2)|θ1, xi2} = lb+l

. Following Vives (2005) [41], we can show that the

best-response function is increasing:

r′ = −dPr{θ2≤ ˆθ2U (x2)|θ1,xi2}

dx2

dPr{θ2≤ ˆθ2U (x2)|θ1,xi2}dxi2

> 0 (22)

Hence, our interest in the existence of monotone equilibria is justified. Although the

problem is now more complicated than for the polar case with n = 1, it is still possible to

show that there exits a unique equilibrium in threshold strategies if the relative precision

of the private information is sufficiently high. Here G(θ∗2U , θ1) is monotonically decreasing

in θ∗2U . But the condition differs from the standard global games setup due to the use of

mixture distributions. The proof is relegated to the Appendix and the result is summarised

22They developed a global game model with mixture distributions at the same time as we did. To ourknowledge both papers are the only papers doing that. However, the focus of Chen et. al. is different toours. They examine the role of rumours in a model of political regime change, while we consider contagionand learning about correlations.

20

in Proposition 3.

Proposition 3 Equilibrium existence, uniqueness and characterization

For a finite precision of the public signal, there exists a finite value γ such that there

exists a unique monotone equilibrium in this sub-game for all γ > γ. Each uninformed

speculator attacks if and only if her private signal is smaller than the threshold x∗2U . A spec-

ulative currency attack is successful if and only if θ2U ≤ θ∗2U .The two equilibrium thresholds

are implicitly defined by the solution to equations (21) and (48).

Proof See Appendix section A.5.3.

Finally, notice that θ∗2U is just a weighted average of the two fundamental equilib-

rium thresholds from the polar case n = 1. As a result: min{θ∗2I,0, θ∗2I,ρH} ≤ θ∗2U ≤

max{θ∗2I,0, θ∗2I,ρH}.

3.3 The novel contagion mechanism

Suppose there was a successful currency attack in the first country, such that the ability of

the government in country 1 to defend its currency must have been low. If fundamentals are

possibly positively correlated across countries, the government’s ability to defend is likely to

be also low in country 2, therefore making a successful currency attack likely to take place in

country 2 as well. However, and perhaps surprisingly, the likelihood of successful currency

attacks can be higher if speculators learn that fundamentals are not correlated (i.e. ρ = 0)

than if speculators do not learn about the correlation.

In particular we demonstrate in this section that the ex-ante likelihood of speculative

attacks when all speculators are informed (n = 1) and learn that fundamentals are uncorre-

lated (i.e. ρ = 0) can be higher than the ex-ante likelihood of attacks when all speculators

are uninformed (n = 0).23 We call this effect contagion-through-alertness, as it arises fol-

lowing a successful currency attack in country 1. Learning good news about the strength

23Notice that ex-ante refers to the beginning of stage 2, that is before θ2 is realised.

21

of a central banks ability to defend its currency might have ”detrimental” effects. This is

because good news can lead to a higher likelihood of crises when it increases the variance

of the posterior distribution relative to the case of not learning any news. The variance

matters despite risk neutrality as knowing what others do is payoff-relevant information in

coordination problems. This effect via the variance of the posterior distribution may lead to

contagion via it’s impact on coordination failure.

The contagion effect can be present for a prior belief that fundamentals are strong and

therefore a large degree of coordination failure. While our result holds more generally, the

special polar cases in which either all speculators are uninformed (n = 0) and all speculators

are informed (n = 1) help to build intuition. What is more than that, our focus on the polar

cases when discussing the contagion case can be sufficient. This will be shown in section

3.5.24 We are interested in uncovering when the ex-ante likelihood of currency attacks is

higher upon learning that fundamentals are uncorrelated, that is when θ∗2I,0 > θ∗2U . From

our discussion of the role of public information and of information precision in section 3.2.2

we learned that there are two effects at work when varying ρH : a mean effect and a variance

effect. These two effects play a key role in what follows.

The mean effect occurs when the informed speculator has a higher posterior mean

relative to the uninformed speculator, which is always true in the case of interest where the

observed fundamentals of country 1 are bad, i.e. θ1 < µ:

E[θ2|xj2] < E[θ2|θ1, xj2] = Pr{ρ = 0|θ1, xj2}αµ+ γxj2α + γ

+ (1− Pr{ρ = 0|θ1, xj2})α

1−ρ2H

[ρHθ1 + (1− ρH)µ] + γxj2α

1−ρ2H

+ γ

Notice that the mean effect works against us becausedθ∗2Udθ1

< 0 for γ sufficiently high. The

mean effect vanishes if θ1 → µ or ρH → 0.

The variance effect refers to a larger variance of the posterior distribution for informed

24When generalising the results of section 3.2.4 to asymmetrically informed speculators and endogenousinformation acquisition in section 3.5.2, we will show how a symmetric equilibrium in country 2 with n∗ = 1can emerge endogenously.

22

speculators relative to uninformed speculators. If the prior belief is that fundamentals are

strong it works in the opposite direction of the mean effect, as it tends to increase θ∗2I,0.

The intuition established in section 3.2.2 goes through. However, the analysis is complicated

because we now have to work with mixture distributions. For that reason it shows to

be more attractive to directly analyse under what conditions the fundamental equilibrium

thresholds satisfy: θ∗2I,0 > θ∗2U . If the latter is the case, then it is due to the variance effect

being sufficiently strong relative to the mean effect. The result is formally summarised in

Proposition 4 below and is derived under the premise that the private signal is sufficiently

precise. Recall that: δ(ρH) ≡ α1−ρ2

H/√

α1−ρ2

H+ γ.

Proposition 4 Existence of the contagion-through-alertness effect

θ∗2I,0 > θ∗2U holds for the prior belief that fundamentals are strong if θ1 ∈ [θ1, µ], where:

θ1 ≡ µ+

(ρHδ(ρH)

((θ∗2 − µ)

[δ(ρH)− α√

α + γ

]+ Φ−1(θ∗2)

[√ γ

α + γ−√

γα

1−ρ2H

+ γ

]))(23)

and θ∗2 solves: ((θ∗2 − µ)

α√α + γ

−√

γ

α + γΦ−1(θ∗2)

)= Φ−1

(l

l + b

)(24)

Proof See Appendix section A.6.

The desired result of θ∗2I,0 > θ∗2U obtains for independent and strong fundamentals if

the variance effect is sufficiently strong relative to the mean effect. Intuitively, the mean

effect is stronger, the lower θ1. As a consequence, the variance effect prevails only if θ1 is not

too small. The contagion-through-alertness effect can only be present for a prior belief that

fundamentals are strong, which implies a large degree of coordination failure. Only here it

can be the case that the right-hand side of equation (23) is negative and, hence, θ1 < µ.

Intuition Contagion-through-alertness can be present even after learning that there is no

exposure. This happens if the higher variance of the posterior distribution ”weighs more”

than the change in the mean of the posterior distribution after good news. At the core of

23

the novel contagion effect is that a higher posterior variance translates into more strategic

uncertainty. Strategic uncertainty refers to the uncertainty about the behaviour of other

speculators as perceived by a given speculator.

In figure 2 we consider a thought experiment that can help us to consolidate the in-

tuition gained so far. We contrast graphically the posterior distributions of informed and

uninformed speculators to illustrate the effect of additional variance of the posterior distri-

bution when the contagion-trough alertness effect exists. Given ρ = 0, informed speculator i

expects a larger fraction of speculators receiving a signal that corresponds to a lower poste-

rior when the other speculators j are informed. Figure 2 sketches this effect as an increase in

the area under the curve left of θ′ for the more dispersed posterior distribution. Therefore, a

larger share of informed speculators than uninformed speculators attack the currency despite

expecting stronger defence of the currency by the government.

0 0.5 1

Qi2I,0=Q'

Qj2I,0ÈQ'

Qj2UÈQ'

Figure 2: More dispersed posterior distribution of informed speculators - more strategicuncertainty

More strategic uncertainty only causes a higher equilibrium likelihood of attacks by

informed speculators if the prior belief is that fundamentals are strong. Then, learning

that fundamentals are uncorrelated reduces the posterior variance and increases strategic

uncertainty. This effect outweighs the mean effect whenever θ1 ∈ (θ1, µ].

24

Numerical example Let us conclude this section with a numerical example. Consider the

following parameters: α =√γ = 2, µ = 0.8, p = 0.5, ρH = 0.7, θ1 = 0.5, l1 = 0.2, b1 = 0.6,

l2 = b2 = 0.5. Notice that the relative cost of attacking is lower in country 1. We find

that θ∗1 ≈ 0.8. As a result speculators in country 2 observe θ1 after a successful speculative

currency attack in country 1, since θ1 < θ∗1. Furthermore, it shows that θ∗2I,0 ≈ 0.31,

θ∗2I,ρH ≈ 0.25 and θ∗2U ≈ 0.29. The likelihood of successful speculative attacks in the state

when ρ = 0 is higher if agents get informed than if they stay uninformed. Notably the effect

is stronger, the higher θ1 (i.e. the weaker the mean effect).

Finally, we have that the likelihood of a spread of the crisis is higher if the cross-country

correlation is positive than if the correlation is zero: Pr{θ2 ≤ θ∗2I,0|ρ = 0} ≈ 0.24 < Pr{θ2 ≤

θ∗2I,ρH |ρ = ρH} ≈ 0.25. However, the latter result does not hold in general because θ∗2I,ρH

decreases in θ1.

3.4 Discussion & relation to empirical literature on contagion

Currency crises show to have a contagious nature. In early empirical work on contagion

Eichengreen et. al. [16] find striking evidence that a crisis elsewhere increases the likelihood

”of a speculative attack by an economically and statistically significant amount” (page 2).

Our theoretical model is consistent with this evidence. In the model the likelihood of suc-

cessful currency attacks in country 2 is higher after country 1 was successfully attacked than

in the scenario where there is no crisis in country 1.25 What is more, this result even holds

if speculators learn that fundamentals are independent, i.e. ρ = 0. Hence, our contagion

mechanism offers a compelling explanation for the abovementioned contagious spread of the

Russian crisis to Brazil, which happened although the interlinkages between the countries

showed to be limited even from an ex-post perspective (compare Bordo and Murshid [6]). In

fact, the likelihood of attacks can be even higher if speculators learn that ρ = 0 than if they

25When θ1 > µ there is no successful currency attack in country 1. Hence, speculators in country 2 do notobserve θ1 and remain uncertain about its realisation. However, speculators in country 2 can infer that therealisation of θ1 must have been sufficiently high, as to prevent a successful attack in country 1. Notice thatfor θ1 > µ the mean and variance effect go in the same direction given a prior belief that fundamentals arestrong. As a result, the likelihood of a successful currency attack in country 2 must be lower when country1 was not successfully attacked.

25

stay uninformed. This is due to an increase in strategic uncertainty caused by the variance

effect. The increased strategic uncertainty is consistent with the view by many ”observers

[who attribute the spread of the Russian crisis to] . . an enhanced perception of risk” (Van

Rijckeghem and Weder [39], p. 294).

The surprising result that the likelihood of successful currency attacks in country 2 may

be higher in the state of the world where ρ = 0 when speculators learn about the correlation

instead of staying uninformed arises if θ1 ∈ (θ1, µ], which implies that θ∗2I,0 > θ∗2U > θ∗2I,ρH . As

a consequence, it may in our model happen that the likelihood of a currency attack is lower

if the cross-country correlation is positive than if the correlation is zero.26 At first glance

this implication is at odds with the existing empirical literature. The empirical literature

prescribes that the likelihood of a spread of the crisis is higher with a positive correlation,

which could be interpreted as a higher institutional similarity or stronger financial and trade

links (compare Dasgupta et. al. [14], Van Rijckeghem and Weder [39] or Glick and Rose [19]).

However, the model implication can potentially offer an explanation why Glick and Rose

found that macroeconomic variables (such as domestic credit, government budget, current

account, international reserves and a devaluation of the real exchange rate) do not help to

explain contagion. When we interpret the cross-country correlation of fundamentals in our

model as reflecting a correlation of macroeconomic variables, then the model suggests that

a positive or zero correlation has an ambiguous effect. Whenever the realisations of θ1 are

relatively high, but still causing a crisis in the ground zero country, the likelihood of a spread

of the crisis may be higher or lower if ρ > 0. Instead if θ1 is low, a positive correlation clearly

increases the likelihood of a spread of the crisis. As a result, the empirical measurement

may not find a significant effect of macroeconomic variables when not accounting for this

non-linearity.

26Although this is only the case if the realisation of θ1 is sufficiently close to µ (see also the numericalexample at the end of section 3.3).

26

3.5 Extension: Asymmetrically informed speculators & endoge-

nous information acquisition

In this section we extend the previous analysis of country 2 to the general case with asym-

metrically informed speculators (0 < n < 1) and demonstrate how endogenous information

acquisition can be triggered by a wake-up call event that makes speculators alert. The game

at date t = 2 has two stages and is solved backwards. First, we solve in section 3.5.1 for the

equilibrium in the second stage, taking n as given. Then we solve the information acquisition

game at the first stage of date t = 2 in section 3.5.2.

Definition 5 A pure strategy Perfect Bayesian Nash Equilibrium in country 2 is an infor-

mation acquisition choice d∗i ∈ {I, U} for each speculator i ∈ [0, 1] in stage 1, an aggregate

fraction of informed speculators n∗ and a decision rule a∗i2d(θ1, xi2, n) in stage 2 such that:

1. All speculators optimally choose di in stage 1 given n∗.

2. The proportion n∗ is consistent with the optimal choices implied by (1.): n∗ =∫ 1

01

{d∗i = I}di.

3. The speculative attack decisions for uninformed speculators in stage 2 are given by:

a∗i2U(θ1, xi2, n∗) = arg max

ai2U∈{0,1}E[u(ai2U , A2, θ2, θ1, n

∗)|xi2] (25)

and for a given realisation of ρ the speculative attack decisions for informed speculators

in stage 2 are given by:

a∗i2I,ρ(θ1, xi2, n∗) = arg max

ai2I∈{0,1}E[u(ai2I , A2, θ2, θ1, ρ, n

∗)|xi2] (26)

4. For a given realisation of ρ the aggregate mass of speculative attackers A2 ≡ A(θ2, n∗, ρ)

in stage 2 is given by:

A(θ2, n∗, ρ) = n∗

∫ +∞

−∞a∗i2I,ρ(θ1, xi2, n

∗)√γφ(√γ(xi2 − θ2))dxi2

+ (1− n∗)∫ +∞

−∞a∗i2U(θ1, xi2, n

∗)√γφ(√γ(xi2 − θ2))dxi2 (27)

27

5. A(θ2, n∗, ρ) is consistent with the optimal speculative attack decision implied by (3.).

3.5.1 Stage 2: The general case 0 < n < 1

Different to before, we now allow for asymmetrically informed speculators. A fraction n of

speculators learns the realisation of the cross-country correlation ρ (informed speculators),

while a fraction 1 − n of speculators does not learn the realisation of the correlation (un-

informed). As before speculators use threshold strategies, where uninformed speculators

attack if their posterior mean is below a threshold. However, differently attack thresholds

now depend on n and for the informed speculators also on the observed correlation. For this

reason we now have three attack thresholds. One critical attack threshold for uninformed

speculators: x∗2U(n). And two critical attack thresholds for informed speculators: x∗2I,ρ(n)

for the two states ρ = 0 and ρ = ρH . Also fundamental thresholds are now functions of n

and we have two of them depending on the realisation of ρ. We denote them with θ∗2,ρ(n) for

the states ρ = 0 and ρ = ρH .

Details on the equilibrium analysis can be found in Appendix section A.7. The equi-

librium can be described by two equations in two unknowns θ∗2,0(n) and θ∗2,ρH (n):

M1(θ∗2,0, θ∗2,ρH

;n) = 0 (28)

M2(θ∗2,0, θ∗2,ρH

;n) = 0 (29)

where n is taken as given. We have that:

∂M1(θ∗2,0, θ∗2,ρH

;n)

∂θ∗2,0> 0 (30)

∂M1(θ∗2,0, θ∗2,ρH

;n)

∂θ∗2,ρH< 0 (31)

From M1(θ∗2,0, θ∗2,ρH

;n) together with equations (30) and (31) we can conclude thatdθ∗2,0dθ∗2,ρH

> 0

for a given n. Furthermore, it shows that∂M2(θ∗2,0,θ

∗2,ρH

;n)

∂θ∗2,0and

∂M2(θ∗2,0,θ∗2,ρH

;n)

∂θ∗2,ρHare negative for

a sufficiently high precision of the private signal γ. Consequently, we can again prove that

there exists a unique equilibrium in threshold strategies for a sufficiently high precision of

28

the private signal. This can be seen by a similar argumentation as in the proof of Proposition

3, using the result thatdθ∗2,0dθ∗2,ρH

> 0 for a given n. The intuition is the same as in the polar

case n = 0 and the result is formally stated in the Proposition 6.

Proposition 6 Equilibrium existence and uniqueness

For a finite precision of the public signal, there exists a finite value γ such that there

exists a unique monotone equilibrium in this sub-game for all γ > γ. Each uninformed

speculator attacks if and only if her private signal is smaller than the threshold x∗2U(n). Each

informed speculator attacks if and only if her private signal is smaller than the threshold

x2I,0(n∗) when learning ρ = 0 and smaller than the threshold x2I,ρH (n∗) when learning ρ = ρH .

A speculative currency attack is successful if and only if θ2 ≤ θ∗2,0(n) (θ2 ≤ θ∗2,ρH (n)) when

ρ = 0 (ρ = ρH).

Proof See Appendix section A.8.

The more interesting question is how a variation in n affects the equilibrium thresholds.

Analytically it is not possible to characterise the equilibrium by using comparative static

methods based on the implicit function theorem for simultaneous equations. In a numeri-

cal analysis we find however very intuitive patterns. Figure 3 shows a numerical example

where parameters are chosen such that the above described contagion mechanism kicks in,

i.e. θ∗2I,0 > θ∗2U(n = 0). Here the likelihood of successful speculative currency attacks shows

to be higher when the actual correlation is ρ = 0 (’good news’) and informed speculators

learn about it, than when speculators remain uninformed. While uninformed speculators

use the same critical attack threshold no matter whether there is a correlation or not, the

informed speculators adjust their critical attack thresholds depending on the observed cor-

relation. Interestingly, the equilibrium fundamental thresholds for the state of the world

when the actual correlation is ρ = 0 and the state of the world when the actual correlation

is ρ = ρH are diverging when n increases. This relations are intuitive. Given that informed

speculators attack more aggressively after learning that ρ = 0 compared to uninformed

speculators, a larger population fraction of informed speculators causes the equilibrium fun-

damental threshold θ∗2,0(n) to be higher (see orange dot-dashed line). The opposite is true

29

for the state of the world, where informed speculators learn that ρ = ρH . Here they attack

less aggressively when compared to uninformed speculators. As a result, the equilibrium

fundamental threshold θ∗2,ρH (n) decreases in n (see green dashed line).

0.0 0.2 0.4 0.6 0.8 1.0n0.21

0.22

0.23

0.24

0.25

0.26

0.27

Θ*2 I,0

Θ*2 I,ΡH

Θ*2 UHn=0LΘ*Ρ=0HnL

Θ*Ρ=ΡHHnL

Figure 3: The critical fundamental thresholds as a function of the fraction of uninformedspeculators n. (Parameters: µ = 0.9, α = γ = 1, b = l = 0.5, P = 0.7, ρH = 0.5 andp = 0.8.)

Analytically, it is difficult to show when the very intuitive first-order effects described

above outweighs potential second-order effects that may arise due to an equilibrium adjust-

ment of the critical attack threshold for uninformed x∗2U(n) when n changes.27

3.5.2 Stage 1: Information acquisition

In the previous section we derived the equilibrium in stage 2 of date 1 for the general case

0 < n < 1. While the amount of information was taken as given – a fraction n ∈ [0, 1] was

informed, we allow for endogenous information acquisition in this section and thereby gener-

alise our result. We argue that there exists an equilibrium in which each speculator acquires

information if the cost of doing so is sufficiently small. The contagion-through-alertness

27An attempt to derive comparative statics results that hold for at least restricted parameter parameterranges using alternative methods is left for future work.

30

effect is present in this equilibrium: there can be more speculative currency attacks after

speculators learn that fundamentals are uncorrelated than without having learned anything.

After observing country 1’s fundamental θ1, speculators in country 2 decide whether

to acquire costly information on the cross-country correlation ρ. Recall that the purchased

information is a perfect signal about the realisation of ρ and that the additional signal is

publicly available to all speculators at a cost. As before, we maintain our focus on the case

in which speculators in country 2 observe a crisis in country 1, that is θ1 < θ∗1 < µ for strong

fundamentals.

The speculator’s problem To determine the equilibrium of the game, we consider the

problem of an individual speculator. Each speculator i takes the population proportion of

speculators n who purchase information as given and compares the expected payoffs from

purchasing the publicly available signal (becoming informed s = I) and not purchasing the

signal (remaining informed s = U). The expected utility of an informed speculator EUI is:

EUI ≡ E[u(d = I, α, γ, µ, ρH , θ1, n)]

= p

( ∫ +∞θ∗2,0(n)

(−l)∫xi2≤x∗2I,0(n)

g(xi2|θ2)dxi2f(θ2)dθ2

+∫ θ∗2,0(n)

−∞ b∫xi2≤x∗2,0(n)

g(xi2|θ2)dxi2f(θ2)dθ2

)(32)

+ (1− p)( ∫ +∞

θ∗2,ρH(n)

(−l)∫xi2≤x∗2,ρ=ρH (θ1,n)

g(xi2|θ2)dxi2f(θ2|θ1, ρH)dθ2

+∫ θ∗2,ρH (n)

−∞ b∫xi2≤x∗2I,ρ=ρH (θ1,n)

g(xi2|θ2)dxi2f(θ2|θ1, ρH)dθ2

)− c

In contrast the expected utility of an uninformed EUU = E[u(d = U, α, γ, µ, ρH , θ1, n)] has

the only difference that the cost c of information is not subtracted and that uninformed

speculators use the same critical attack threshold x∗2U(n) for both states of the word. The

distributions of the fundamental in country 2 for both states of the world and the distribution

31

of signals are given as follows:

f(θ2) =

√α

2πexp{−

α2

(θ2−µ))2} (33)

f(θ2|θ1, ρH) =

√α

2π(1− ρ2H)

exp{− α

2(1−ρ2H

)(θ2−(ρHθ1+(1−ρH)µ))2}

(34)

g(x|θ2) =

√γ

2πexp{−

γ2

(x−θ2)2} (35)

Intuition Before the fundamental θ2 is realised, speculators know the conditional distribu-

tion of θ2, which depends on the state of the world. For informed speculators receiving news

that fundamentals are uncorrelated (correlated), the pdf is given by f(θ2) ( f(θ2|θ1, ρH) ).

The difference in the expected payoffs of informed and uninformed speculators results from

the informed being able to select different critical attack threshold for the two events. For

each realisation of θ2, speculators can compute how many (un-) informed speculators decide

to attack and how likely it is that they themselves receive a private signal below their critical

threshold which induces them to attack. Both, informed and uninformed speculators know

that the two events ”no correlation” and ”positive correlation” occur with probability p and

1− p, respectively. For each event speculators integrate over the corresponding distribution

of θ2.

Benefits from and costs of attacking To gain a better understanding consider the

benefits and costs from attacking for the polar case when n = 0. Here θ∗2,0(n = 0) =

θ∗2,rhoH (n = 0) = θ∗2U . Taking derivatives leads to:

dEU

dx∗2I,0(0)= p

( −l ∫ +∞θ∗2U

g(x∗2I,0(0)|θ2)f(θ2)dθ2

+b∫ θ∗2U−∞ g(x∗2I,0(0)|θ2)f(θ2)dθ2

)

The first summand is negative and represents the cost of increasing the attack threshold

due to a higher likelihood to participate in unsuccessful speculative attacks. The second

summand is positive and represents the benefit from a higher likelihood to participate in

successful currency attacks. In equilibrium the marginal cost and the marginal benefit have

to be equalised.

32

Strategic complementarity in information acquisition choices A given speculator

finds it optimal to purchase the publicly available signal if the expected differential payoff is

positive. If the dependency of equilibrium fundamental thresholds can be characterised as

in figure 3,28 then we have a strategic complementarity in information acquisition choices.

Here we have that incentives to get informed are increasing in n.

When is it optimal to purchase information? If the differential expected payoff EUI−

EUU ≡ ∆[α, γ, µ, ρH , θ1, n] is positive, which can be written as:

p

( ∫ +∞θ∗2,0(n)

(−l)∫ x∗2I,0(n)

x∗2U (n) g(xi2|θ2)dxi2f(θ2)dθ2

+∫ θ∗2,0(n)

−∞ b∫ x∗2I,0(n)

x∗2U (n) g(xi2|θ2)dxi2f(θ2)dθ2

)−

(1− p)( ∫ +∞

θ∗2,ρH(n)

(−l)∫ x∗2U (n)

x∗2I,ρH(n)g(xi2|θ2)dxi2f(θ2|θ1, ρH)dθ2

+∫ θ∗2,ρH (n)

−∞ b∫ x∗2U (n)

x∗2I,ρH(n)g(xi2|θ2)dxi2f(θ2|θ1, ρH)dθ2

)− c ≥ 0 (36)

Suppose we are in the scenario where the novel contagion effect occurs, i.e. θ∗2,0(n) ≥

θ∗2,ρH (n). Given that an increase in n is associated with an increase in θ∗2,0(n) and a decrease

in θ∗2,ρH (n). An increase in n leads to a relative increase of the benefit component in the first

summand and a relative decrease of the loss component in the second summand, holding

everything else equal. For any admissible combination of equilibrium attack thresholds this

implies a strict increase in the differential payoff of being informed. The reason being that

informed speculators can take ”full” advantage of the change in equilibrium fundamental

thresholds when n changes, while uninformed speculators have always to ”balance” the

marginal benefit from increasing x∗2U in case there is no exposure (with probability p) with

the marginal loss of increasing x∗2U in case there is an exposure (with probability 1− p).

A consequence of the above argument is that if the cost of information is sufficiently

low as to give an incentive for an individual speculator to acquire information given that all

other speculators are uninformed (i.e. n = 0), then it is also optional to acquire information

for an individual speculator no matter how many other speculators are informed (i.e. for all

28That is if θ∗2,0(n) > θ∗2U > θ∗2,ρH (n) and if θ∗2,0(n) is monotonically increasing in n, while θ∗2,ρH (n) ismonotonically decreasing in n. Notice that the former implies that x∗2U (n) ∈ (x∗2I,ρH (n), x∗2I,0(n)) for alln ∈ (0, 1].

33

n ∈ (0, 1]). The result is summarised below.

Result. Equilibrium of the information acquisition game

Suppose that speculators have a prior belief that fundamentals are strong, and that private

signals are sufficiently precise such that there exists a unique monotone equilibrium of the

sub-game at stage 2 for a given n. Then there exists a unique equilibrium of the information

acquisition game at stage 1 in which all speculators acquire the publicly available signal, i.e.

n∗ = 1, after observing θ1 < µ whenever:

1. ∆[α, γ, µ, ρH , θ1, n = 0] > 0

2. there is a strategic complementarity in information acquisition choices.

The strategic complementarity in information acquisition choices is guaranteed if parameters

are such that θ∗2,0(n) > θ∗2U > θ∗2,ρH (n) and that θ∗2,0(n) is monotonically increasing in n,

while θ∗2,ρH (n) is monotonically decreasing in n.

3.5.3 Discussion

In this section we demonstrate that the contagion-through-alertness effect described

earlier can be an equilibrium phenomenon in the more general setup with endogenous in-

formation acquisition whenever the cost of information is sufficiently low. Furthermore, we

found that we can have a strategic complementarity in information acquisition choices. The

strategic complementarity in information acquisition choices arises quite naturally in global

games models with endogenous information acquisition.29

The numerical example underlying figure 3 provides a situation when the above re-

sult applies. Here, we have that θ∗2,0(n) > θ∗2U > θ∗2,ρH (n) and that θ∗2,0(n) is monotonically

increasing in n, while θ∗2,ρH (n) is monotonically decreasing in n. Although this characterisa-

tion suggest to hold generally in our numerical analysis, it is not possible to do an analytical

comparative statics analysis relying on the simultaneous equations version of the implicit

29Szkup and Trevino [38] show numerically in a model with continuous information acquisition choice overthe precision of private signals and convex costs that strategic complementarity may under some parametersnot be guaranteed. However, in our model with discrete information acquisition choice and publicly availablesignals their result should be less or not at all relevant.

34

function theorem. The problem is left for future research.

Policy implications A wake-up call triggers endogenous information acquisition whenever

the cost of information is sufficiently low. The benefit from being informed shows to be

positively related to the difference between θ∗2,0(n) and θ∗2,ρH (n). As a result, the incentives

to get informed are the higher, the stronger the contagion mechanism.

If θ1 ∈ (θ1, µ], then the contagion-through-alertness effect prevails and we have a higher

likelihood of successful currency attacks after speculators learn that there is no correlation

compared with the case where they stay uninformed. If instead θ1 < θ1 , then the we have a

higher likelihood of successful currency attacks after speculators learn that there is a positive

correlation compared with the case where they stay uninformed. In both scenarios, an

informed policy maker could reduce the likelihood of successful currency attacks by making

information more costly, such that individual speculators optimally decide not to acquire

information in the first place.

The opposite is true if θ1 ∈ (θ1, µ] and informed speculators learn that there is exposure

or if θ1 < θ1 and informed speculators learn that there is no exposure. Here, an informed

policy maker could reduce the likelihood of successful currency attacks by making information

less costly, such that individual speculators optimally decide to acquire information.

4 Related literature

The literature on currency crisis is large and we do not attempt to provide a detailed re-

view but focus on the incomplete information game introduced by Morris and Shin [29, 30].

Following the seminal contribution of Carlsson and van Damme [8], a perturbation of the

information structure yields a unique equilibrium. This overcomes the multiplicity of equilib-

ria present in many previous models of currency crisis, such as the Krugman-Flood-Garber

[24, 17] first-generation currency crisis model, the second-generation currency crisis model

by Obstfeld [32], and many third-generation currency crisis models.

An important ingredient of our contagion-through-alertness mechanism is the exac-

35

erbation of the coordination problem when the precision of the agents’ prior information

changes. This element is present in earlier work on bank runs by Rochet and Vives [36], for

example. Our contagion mechanism sheds new light on results on the role of information

precision and of public information that have been established in the global games litera-

ture. The novelty of this paper is to combination the mean effect and the variance effect in

a setting where both can go in opposite directions.

Furthermore, our paper is also related to the literature analysing the role of information

precision. Information acquisition can have a detrimental effect in our model. This result

connects to papers that stress the possible benefits of coarse information.30 For instance, the

papers of Dang, Gorton and Holmstrom [11] as well as Pagano and Volpin [33] emphasise

the benefits of coarse information in supporting market liquidity.

Endogenous information acquisition is considered by Hellwig and Veldkamp [22] who

discuss the similarity in the strategic motives between choosing an action and deciding on

how much information to acquire in a beauty-contest model. In their words, investors “who

want to do what others do, want to know what others know” (p. 223). They show that

that adding a public information choice may lead to a multiplicity in equilibria. By contrast,

uniqueness is always guaranteed under the usual mild condition of sufficiently precise private

signals in our global games model.

Contagion in financial economics

While there exists a large literature on financial contagion, typically either interconnect-

edness or common exposures is required to generate contagion or systemic fragility more

generally. First, systemic fragility because of common exposures (correlated fundamentals)

are considered in Acharya and Yorulmazer [1], who show that banks can have an ex-ante

incentive to correlate their investment decision to avoid information contagion, and Allen,

Babus and Carletti [3], who analyze systemic risk resulting from the interaction of common

exposures and funding maturity through an information channel. Manz [26] explores the

role of common exposures in a global-games framework. Second, financial contagion can

30See Morris and Shin [31].

36

arise from interconnectedness. Allen and Gale [4] provide a model of financial contagion

as an equilibrium outcome through interbank linkages. Dasgupta [12] shows that financial

contagion arises with positive probability in a global-game version of Allen and Gale [4]. In

Goldstein and Pauzner [20] contagion results from a wealth effect of investors who become

more averse to strategic risk after a crisis in one country. There is also a large literature

on contagion through a pecuniary “fire-sale” externality related to the ideas of Shleifer and

Vishny [37].

The distinct feature of the proposed contagion-through-alertness mechanism is the

endogenous information acquisition such that contagion can occur in the absence of inter-

connectedness and common exposures. Observing an adverse event in another region is a

wake-up call to investors that induces them to acquire costly information about their expo-

sure to that event. This alertness effect can result in a higher likelihood of an adverse event

in their region. Such fragility can even be present if investors learn that their investments are

completely uncorrelated with the adverse event. In sum, it is sufficient that fundamentals

are potentially correlated to generate the alertness effect. Once speculators are alert, the

incidence of speculative attacks is increased even after speculators learn that the regional

fundamentals are uncorrelated.

Contagion in international finance

The international finance literature mainly considers a terms-of-trade channel and a common-

discount-factor channel to explain an international co-movement in asset prices during crisis

periods. (Co-movement in asset prices is considered as contagious when “excessive”). How-

ever, these channels cannot account for the observed co-movements in the 1997/1998 emerg-

ing market crisis period. Pavlova and Rigobon [34] argue that neither channel explains the

co-movements in asset prices of countries with limited trade links. They construct an open-

economy dynamic stochastic general equilibrium model and show that portfolio constraints

can cause a substantial amplification and help to explain the observed co-movements in asset

prices in crisis periods. An alternative amplification mechanism is provided by Kodres and

Pritsker [23] who establish the “cross-market portfolio rebalancing channel”, which is based

37

on the common discount factor channel.

Calvo and Mendoza [7] also offer a contagion mechanism that does not rely on cor-

related macroeconomic fundamentals, where the authors relate contagion to information

acquisition. In this sense their paper is closer to our model than the existing mechanisms in

the financial economics literature. In their paper a lower degree of information acquisition,

as a consequence of globalisation, gives rise to contagion because market participants prefer

imitate arbitrary market portfolios instead of gathering information which can lead to a

detrimental herding behaviour. By contrast, contagion is a consequence of a higher, not a

lower, degree of information acquisition in our model, where fragility can arise because of

heightened strategic uncertainty in coordination problems.

5 Conclusion

This paper proposes a novel contagion mechanism based on an alertness effect. Upon observ-

ing a successful currency attack elsewhere – a wake-up call – speculators wish to determine

to what extent their investment position is affected by that crisis. This alertness effect per se

can lead to a larger likelihood of successful currency attacks through an increase in strategic

uncertainty. The contagion-through-alertness effect prevails whenever speculators in coun-

try 2 observe a successful currency attack in country 1 that results from weak but not too

weak fundamentals. We consider this scenario as relevant. First, a very low fundamental

realisation of θ1 is a low probability event. Second, the situation in practise is most of the

time not so obvious and fragile countries or banks tend to be somewhat weak but not des-

tined to fail with certainty. While we present an application to speculative currency attacks,

the contagion-through-alertness mechanism occurs in general coordination problems and is

applicable to bank runs, political regime change, and sovereign debt crises.

38

A Appendix

A.1 Country 1

A.1.1 Equilibrium analysis

The first equilibrium condition is given by:

A(θ1) = Pr{xi1 ≤ x∗1|θ∗1} = Φ(√

γ(x∗1 − θ1))

= θ∗1

x∗1 = θ∗1 +1√γ

Φ−1(θ∗1) (37)

It demands that in equilibrium the critical fraction of attacking speculators has to be equal

to the critical fundamental threshold above which it pays to act.

The second equilibrium condition is an indifference condition. It implicitly defines the

equilibrium fundamental threshold. Given θ∗1, the payoff of an attacking speculator is given

by:

bPr{θ1 ≤ θ∗1|xi1} − lPr{θ1 > θ∗1|xi1} = 0 (38)

where:

Pr{θ1 ≤ θ∗1|xi1} = Φ(θ∗1 − E[θ1|xi1]√

Var[θ1|xi1]

)= Φ

(√α + γ(θ∗1 −

αµ+ γxi1α + γ

))

which is decreasing in x∗1. A speculator attacks if and only if xi1 ≤ x∗1. At the critical

equilibrium attack threshold x∗1 speculators have to be just indifferent whether to attack or

not.

Combining the two equilibrium conditions leads to equation (6). The right-hand side

is a constant and the left-hand side is decreasing in θ1 if the relative precision of the private

signal is sufficiently high:

dF1(θ1)

dθ1

= Φ′ ∗α−√γ 1

φ(Φ−1(θ1))√α + γ

< 0 ifα√γ<√

2π (39)

39

A.1.2 Equilibrium characterization

It is useful to distinguish between a prior belief that fundamentals are strong and a prior

belief that fundamentals are weak.

Consider the equilibrium condition:

Φ

(α√α + γ

(θ∗1 − µ)−√

γ

α + γΦ−1(θ∗1)

)=

l

b+ l< 1 (40)

Reformulate it to:

α(θ∗1 − µ) =√γΦ−1(θ∗1) +

√α + γΦ−1

( l

b+ l

)(41)

and notice that, given µ ∈ (0, 1), θ∗1 = µ if and only if:

√γΦ−1(µ) +

√α + γΦ−1

( l

b+ l

)= 0 (42)

θ∗1 > µ if and only if:√γΦ−1(µ) +

√α + γΦ−1

( l

b+ l

)< 0 (43)

and θ∗1 < µ otherwise.

Equation (43) refers to the case of a prior belief that fundamentals are ”weak”. Weak

fundamentals are associated with a low µ and a relatively low cost of an unsuccessful currency

attack. Here the critical equilibrium fundamental threshold is strictly larger than µ. The

opposite is true if the prior belief is that fundamentals are ”strong”, meaning that µ is high

and the relative cost of an unsuccessful attack is high. Of special interest is the case when

lb+l

= 12

for which the analysis simplifies. Here a prior belief that fundamentals are weak

(strong) is defined as 0 < µ < 12

(12< µ < 1). For a prior belief that fundamentals are weak

(strong) we can find that: 0 < µ < 12< θ∗1 < x∗1 < 1 (0 < x∗1 < θ∗1 <

12< µ < 1).

40

A.2 Country 2: Stage 2

A.2.1 Higher precision of the public signal (α) and the private signal (γ)

The subsequent discussion draws in parts from Bannier and Heinemann [5]. When analysing

the equilibrium condition we find that:

dθ∗2I,ρdα

< 0 if θ∗2I,ρ < µ′2(ρ, θ1) + 1

2√

α1−ρ2

+γΦ−1

(lb+l

)≥ 0 otherwise.

and:

dθ∗2I,ρdγ

> 0 if θ∗2I,ρ < µ′2(ρ, θ1) + 1√

α1−ρ2

+γΦ−1

(lb+l

)≤ 0 otherwise.

If lb+l≥ 1

2, then a prior belief that fundamentals are strong (i.e. θ∗2I,ρ < µ′2(ρ, θ1) for

ρ = 0 and ρ = ρH) implies thatdθ∗2I,ρdα

< 0 anddθ∗2I,ρdγ

> 0. And if lb+l

< 12, then a prior

belief that fundamentals are weak (i.e. θ∗2I,ρ > µ′2(ρ, θ1) for ρ = 0 and ρ = ρH) implies thatdθ∗2I,ρdα

> 0 anddθ∗2I,ρdγ

< 0.

Moreover, if lb+l

< 12, then the prior belief that fundamentals are strong does not

necessarily imply thatdθ∗2I,ρdα

< 0 anddθ∗2I,ρdγ

> 0. This is only true if µ′2(ρ, θ1) is sufficiently

high. The critical values of µ′2(ρ, θ1) can be derived from the equilibrium condition after

plugging in the above inequalities. For instance we find that if lb+l

< 12, then

dθ∗2I,ρdα

< 0 when

µ′2(ρ, θ1) ≥ [ρθ1 + (1− ρ)µ] where:

µ ≡ Φ

((α√γ

1

2√

α1−ρ2 + γ

−

√α

1−ρ2 + γ√γ

)Φ−1

(l

l + b

))− 1

2√

α1−ρ2 + γ

Φ−1

(l

l + b

). (44)

Similarly, if lb+l≥ 1

2, then the prior belief that fundamentals are weak does not nec-

essarily imply thatdθ∗2I,ρdα

> 0 anddθ∗2I,ρdγ

< 0. For instance we only have thatdθ∗2I,ρdα

> 0, if

µ′2(ρ, θ1) is sufficiently low, i.e. if µ′2(ρ, θ1) ≤ [ρθ1 + (1− ρ)µ].

41

A.3 The special case n = 1

In this paragraph we provide details for the equilibrium analysis in section 3.2.1. The first

equilibrium condition is identical to equation (38) with the only difference that θ∗1 needs to

be substituted by θ∗2 . Instead the second equilibrium condition, which is an indifference

condition, can be computed as:

bPr{θ2 ≤ θ∗2|x∗2} − lPr{θ2 > θ∗2|x∗2} = 0 (45)

where:

Pr{θ2 ≤ θ∗2|x2} = Φ

(√α

1− ρ2+ γ(θ∗2 −

α1−ρ2

α1−ρ2 + γ

[ρθ1 + (1− ρ)µ]− γα

1−ρ2 + γx∗2))

Equation (10) is constructed by combining both equilibrium conditions. Using the same

argument as before, it can be shown that equation (10) has a unique solution if the relative

precision of the private signal is sufficiently strong, i.e. ifα

1−ρ2H√γ<√

2π.

A.4 Bayesian updating

In this section we analyse how the posterior probability of facing the state ρ = 0 varies with

the private signal. Differentiating equation (16) with respect to xi2 leads to:

dPr{ρ = 0|θ1, xi2}dxi2

(46)

=

p(1− p)

(

1α

+ 1γ

)−1(√1−ρ2

H

α+ 1

γ

)−1

φ′(

xi2−µ√1α

+ 1γ

)φ

(xi2−[ρHθ1+(1−ρH)µ]√

1−ρ2H

α+ 1γ

)−(√

1α

+ 1γ

)−1(1−ρ2

H

α+ 1

γ

)−1

φ

(xi2−µ√

1α

+ 1γ

)φ′(xi2−[ρHθ1+(1−ρH)µ]√

1−ρ2H

α+ 1γ

)

[p

(√1α

+ 1γ

)−1

φ

(xi2−µ√

1α

+ 1γ

)+ (1− p)

(√1−ρ2

H

α+ 1

γ

)−1

φ

(xi2−[ρHθ1+(1−ρH)µ]√

1−ρ2H

α+ 1γ

)]2

42

To determine the sign, we have to inspect the nominator of equation (46). After several

manipulations, we find that the nominator is weakly positive if:

(α + γ(1− ρ2

H)

α + γ

)(µ− xi2) > [ρHθ1 + (1− ρH)µ]− xi2 (47)

and negative otherwise. Hence, we arrive at the result summarised in equation (20) and the

discussion thereafter.

A.5 Derivations related to the equilibrium analysis for the special

case n = 0

A.5.1 The equilibrium condition

In this paragraph we provide details for the equilibrium analysis in section 3.2.4. Again two

equilibrium conditions have to be satisfied. First, the critical fraction of attacking speculators

A(θ∗2U) has to equal the critical fundamental threshold above which it pays to attack. This

leads to the first equilibrium condition:

x∗2U = θ∗2U +

√1

γΦ−1(θ∗2U) (48)

where the subscript U stands for uninformed. Second, a speculator with the threshold signal

x∗2U has to be indifferent whether to attack the currency or not given θ∗2U :

bPr{θ2 ≤ θ∗2U |θ1, x∗2U} − lPr{θ2 > θ∗2U |θ1, x

∗2U} = 0 (49)

where:

Pr{θ2 ≤ θ∗2U |θ1, x∗2U} = Pr{s = NE|θ1, x

∗2U}Φ

(θ∗2U −

αµ+γx∗2Uα+γ√

1α+γ

)

+ Pr{s = E|θ1, x∗2U}Φ

(θ∗2U − α

1−ρ2H

[ρHθ1+(1−ρH)µ]+γx∗2Uα

1−ρ2H

+γ√1

α

1−ρ2H

+γ

)

43

The indifference condition shows to be a mixture between the indifference conditions

for the two cases n = 1, ρ = 0 and n = 1, ρ = ρH . A combination of equations (48) and (49)

leads to the equilibrium condition stated in equation (21).

A.5.2 Monotonicity

The conditional density function f (x|θ) is normal with mean θ and satisfies the monotone

likelihood ratio property (MLRP). For all xi > xj and θ′ > θ, we have:

f (xi|θ′)f (xi|θ)

≥ f (xj|θ′)f (xj|θ)

or:φ(√

γ (xi − θ′))

φ(√

γ (xi − θ)) ≥ φ

(√γ (xj − θ′)

)φ(√

γ (xj − θ)) .

As a consequence, we can make us of the result in Proposition 1 of Milgrom [28] and conclude

that Pr {θ2 ≤ θ∗2U |θ1, xi2} is monotonically decreasing in xi2.

Furthermore, notice thatdPr{θ2≤θ∗2U |θ1,x2}

dθ∗2U> 0 and from equation (48) we can derive

that:

0 ≤ dθ2(x2)

dx2

≤ 1

1 +√

2πγ

. (50)

A.5.3 Proof of Proposition 3

The result in Proposition 3 can be proven by showing that dG(θ2U ,θ1)dθ2

< 0 for some sufficiently

high value of γ. We have that:

dG(θ2U , θ1)

dθ2

= Pr{ρ = 0|θ1, x2U(θ2)}dΦI,ρ=0

dθ2

+ (1− Pr{ρ = 0|θ1, x2U(θ2)})dΦII,ρ=ρH

dθ2

+dPr{ρ = 0|θ1, x2U(θ2)}

dx2U

dx2U(θ2)

dθ2

[ΦI,ρ=0 − ΦII,ρ=0

](51)

The proof proceeds by inspecting the individual terms of equation (51). From our earlier

analysis we know thatdΦI,ρ=0

dθ2< 0 if α√

γ<√

2π and thatdΦII,ρ=ρH

dθ2< 0 if

α

1−ρ2H√γ<√

2π. Notice

that limγ→∞dΦI,ρ=0

dθ2= limγ→∞

dΦII,ρ=ρHdθ2

= −1.

44

The sign of the last summand in equation (51) is ambiguous. We have that[ΦI,ρ=0 −

ΦII,ρ=ρH

]≤ 0 whenever θ∗2I,0 ≤ θ∗2I,ρH and

[ΦI,ρ=0−ΦII,ρ=ρH

]> 0 otherwise. Furthermore, it

shows that limγ→∞[ΦI,ρ=0−ΦII,ρ=ρH

]= 0. The last term to consider is dPr{ρ=0|θ1,x2U (θ2)}

dx2U (θ2)dx2U

dθ2.

Given the previous sufficient conditions on the relative precision of the private signal we have

that:

0 <dx2U

dθ2

= 1 +1√γ

1

φ(Φ−1(θ2))< 1 +

√2π

α

Finally, recall dPr{ρ=0|θ1,xi2}dxi2

from section 3.2.3. Taking the limit γ →∞ shows that dPr{ρ=0|θ1,x2U}dx2U

is finite as long as α is finite. Hence, we can conclude that:

limγ→∞

dPr{ρ = 0|θ1, x2U(θ2)}dx2U

dx2U(θ2)

dθ2

[ΦI − ΦII

]= 0

As a result, there must exist a finite level of precision γ such that dG(θ2U ,θ1)dθ2

< 0 for all

γ > γ, as long as α takes on a finite value. This concludes the proof of Proposition 3.

A.6 Proof of Proposition 4

The result in Proposition 4 is proven by analysing the equilibrium condition for n = 0. First,

recall that θ∗2U solves equation (21). Both ΦI,ρ=0(θ∗2U) and ΦI,ρ=ρH (θ∗2U , θ1) are decreasing in

θ∗2U if α√γ<√

2π. Second, consider the equilibrium condition for the polar case n = 1 and

observe that it can only be true that θ∗2I,0 > θ∗2U if ΦI,ρ=0(θ∗2) > ΦI,ρ=ρH (θ∗2, θ1). The condition

in equation (23) follows immediately after few manipulations. Since we are interested in a

condition such that θ∗2I,0 > θ∗2U , equation (23) has to be evaluated at θ∗2I,0. This explains

equation (24) and completes the proof.

A.7 Derivations related to the equilibrium analysis for the general

case 0 < n < 1

The equilibrium conditions can again be derived in two steps. First, in equilibrium the

fraction of attacking speculators A2(θ∗2,ρ) has to be equal to the fundamental threshold θ∗2,ρ(n)

45

above which it pays to act. This leads to two equilibrium conditions:

θ∗2,0(n) = nΦ

(x∗2I,0(n)− θ∗2,0(n)

1√γ

)+ (1− n)Φ

(x∗2U(n)− θ∗2,0(n)

1√γ

)(52)

θ∗2,ρH (n) = nΦ

(x∗2I,ρH (n)− θ∗2,ρH (n)

1√γ

)+ (1− n)Φ

(x∗2U(n)− θ∗2,ρH (n)

1√γ

)(53)

Second, in equilibrium the speculator receiving a private signal equal to the equilib-

rium threshold has to be indifferent whether to attack or not. This has to hold for both,

uninformed speculators and informed speculators who learn that ρ = 0 or ρ = ρH . We arrive

at three equilibrium conditions. One equilibrium condition for uninformed speculators:

J(θ∗2,0, θ∗2,ρH

, x∗2U ;n)

≡ Pr{ρ = 0|θ1, x∗2U(n)}ΦJ,0(θ∗2,0(n), x∗2U(n))

+ (1− Pr{ρ = 0|θ1, x∗2U(n)})ΦJ,ρH (θ∗2,ρH (n), x∗2U(n)) =

l

b+ l(54)

where:

ΦJ,0(θ∗2,0(n), x∗2U(n)) ≡ Φ

(θ∗2,0(n)− αµ+γx∗2U (n)

α+γ√1

α+γ

)(55)

ΦJ,ρH (θ∗2,ρH (n), x∗2U(n)) ≡ Φ

(θ∗2,ρH (n)−α

1−ρH[ρHθ1+(1−ρH)µ]+γx∗2U (n)

α

1−ρ2H

+γ√1

α

1−ρ2H

+γ

)(56)

And two equilibrium conditions for informed speculators:

Φ

(θ∗2,0(n)− αµ+γx∗2I,0(n)

α+γ√1

α+γ

)=

l

b+ l(57)

46

and:

Φ

(θ∗2,ρH (n)−α

1−ρ2H

[ρHθ1+(1−ρH)µ]+γx∗2I,ρH(n)

α

1−ρ2H

+γ√1

α

1−ρ2H

+γ

)=

l

b+ l(58)

We are left with five equation in five unknowns. First, we can use equation (52) to

obtain x∗2U(n) as a function of θ∗2,0(n), and x∗2I,0(n). Second, we can use equation (57) to

obtain x∗2I,0(n) as a function of θ∗2,0(n).

Plugging the second function into the first function leads to:

x∗2U(θ∗2,0;n) = θ∗2,0 (59)

+

√1

γΦ−1

(θ∗2,0 − nΦ(α(θ∗2,0−µ)−

√α+γΦ−1( l

b+l)

√γ

)1− n

)

Notice that:

∂x∗2U(θ∗2,0;n)

∂θ∗2,0= 1 +

√1

γ

1−nΦ′

(α(θ∗2,0−µ)−

√α+γΦ−1( l

b+l)

√γ

)α√γ

1−n

φ

(Φ−1

( θ∗2,0−nΦ

(α(θ∗2,0−µ)−

√α+γΦ−1( l

b+l)

√γ

)1−n

)) > 0 (60)

when assuming that the sufficient condition for uniqueness from the polar case n = 1

holds, i.e. α√γ<√

2π. In equilibrium there can only be one critical threshold for unin-

formed speculators, x∗2U(θ∗2,0;n) = x∗2U(n). Hence, equation (59) can in turn be plugged into

ΦJ,0(θ∗2,0(n), x∗2U(n)), which gives us ΦJ,0 as a function of θ∗2,0(n) only.

Similarly, we can use equation (53) to obtain x∗2U(n) as a function of θ∗2,ρH (n) and

x∗2I,ρH (n). Then we can use equation (58) to obtain x∗2I,ρH (n) as a function of θ∗2,ρH (n).

47

Again plugging the second into the first function leads to:

x∗2U(θ∗2,ρH ;n) = θ∗2,ρH (61)

+

√1

γΦ−1

(θ∗2,ρH − nΦ( α

1−ρ2H

(θ∗2,ρH−[ρHθ1+(1−ρH)µ])−

√α

1−ρH+γΦ−1( l

b+l)

√γ

)1− n

)

Analog to before we have that:

∂x∗2U(θ∗2,ρH ;n)

∂θ∗2,ρH= 1+

√1

γ

1−nΦ′

(α

1−ρ2H

(θ∗2,ρH−[ρHθ1+(1−ρH )µ])−

√α

1−ρH+γΦ−1( l

b+l)

√γ

)α

(1−ρ2H

)√γ

1−n

φ

(Φ−1

( θ∗2,ρH−nΦ

(α

1−ρ2H

(θ∗2,ρH−[ρHθ1+(1−ρH )µ])−

√α

1−ρH+γΦ−1( l

b+l)

√γ

)1−n

)) > 0

given the sufficient condition for uniqueness from the polar case n = 1 holds, i.e.α

1−ρ2H√γ

=√

2π. Again we can use the argument that in equilibrium there can only be one critical

threshold for uninformed speculators, x∗2U(θ∗2,ρH ;n) = x∗2U(n). Hence, plugging equation (61)

into ΦJ,ρH (θ∗2,ρH (n), x∗2U(n)) gives us ΦJ,ρH as a function of θ∗2,ρH (n) only.

Equalising equations (59) and (61) gives an implicit relation between θ∗2,0(n) and

θ∗2,ρH (n):

M1(θ∗2,0, θ∗2,0;n) ≡ x∗2U(θ∗2,0;n)− x∗2U(θ∗2,ρH ;n) = 0 (62)

Finally, consider J(θ∗2,0(n), θ∗2,ρ=H(n), x∗2U(n)) and plug in for x∗2U(n) from equation (59).

Let us define:

M2(θ∗2,ρ=0, θ∗2,ρH

;n) ≡ J(θ∗2,0, θ∗2,ρ=H ;n)− l

b+ l= 0 (63)

where n is taken as given.

48

We can derive:

∂M1(θ∗2,0, θ∗2,ρH

;n)

∂n=

< 0 if θ∗2I,0 > θ∗2I,ρH

≥ 0 if θ∗2I,0 ≤ θ∗2I,ρH

Q 0 otherwise.

(64)

∂M2(θ∗2,0, θ∗2,ρH

;n)

∂θ∗2,0=

dPr{ρ = 0|θ1, x∗2U(θ∗2,0;n))}

dθ∗2,0[ΦJ,0 − ΦJ,ρH ]

+ Pr{ρ = 0|θ1, x∗2U(θ∗2,0;n))}

dΦJ1(θ∗2,0;n)

dθ∗2,0(65)

∂M2(θ∗2,0, θ∗2,ρH

;n)

∂θ∗2,ρH= (1− Pr{ρ = 0|θ1, x

∗2U(θ∗2,0;n)})

dΦJ,ρH (θ∗2,ρH ;n)

dθ∗2,ρH(66)

A.8 Proof of Proposition 6

The proof of Proposition 6 is similar to the proof of Proposition 3. We consider equations

(66) and (65) in turn.

First, observe thatdM2(θ∗2,0,θ

∗2,ρH

;n)

dθ∗2,ρH< 0 is satisfied if:

α1−ρ2

H√γ<

1−nΦ′

(α

1−ρ2H

(θ∗2,ρH−[ρHθ1+(1−ρH )µ])−

√α

1−ρH+γΦ−1( l

b+l)

√γ

)α

(1−ρ2H

)√γ

1−n

φ

(Φ−1

( θ∗2,ρH−nΦ

(α

1−ρ2H

(θ∗2,ρH−[ρHθ1+(1−ρH )µ])−

√α

1−ρH+γΦ−1( l

b+l)

√γ

)1−n

))

Notice that the standard normal pdf cannot take values above 1√2π

. As a result the above

equation holds given the sufficient condition used for the polar case n = 1, i.e. α√γ<√

2π.

Second, observe thatdM2(θ∗2,0,θ

∗2,ρH

;n)

dθ∗2,0< 0 is satisfied for a sufficiently high but finite

γ (given a finite α). This can be seen by applying the same argument as in the proof of

Proposition 3.

Finally, recall thatdθ∗2,0dθ∗2,ρH

> 0 for a given n. This connects the two results above and lets

us conclude that the left-hand side of M2(θ∗2,0, θ∗2,ρH

;n) is strictly decreasing in θ∗2,0 whenever

γ is sufficiently high (given a finite α). As the right-hand side is constant, this concludes our

49

proof that there must exist a γ such there does exist a unique monotone equilibrium for all

γ > γ (given a finite α).

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